1 / 56

Cross Tabulation and Chi Square Test for Independence

Cross Tabulation and Chi Square Test for Independence. Cross-tabulation. Helps answer questions about whether two or more variables of interest are linked: Is the type of mouthwash user (heavy or light) related to gender?

teneil
Download Presentation

Cross Tabulation and Chi Square Test for Independence

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Cross Tabulation and Chi SquareTest for Independence

  2. Cross-tabulation • Helps answer questions about whether two or more variables of interest are linked: • Is the type of mouthwash user (heavy or light) related to gender? • Is the preference for a certain flavor (cherry or lemon) related to the geographic region (north, south, east, west)? • Is income level associated with gender? • Cross-tabulation determines association not causality.

  3. Dependent and Independent Variables • The variable being studied is called the dependent variable or response variable. • A variable that influences the dependent variable is called independent variable.

  4. Cross-tabulation • Cross-tabulation of two or more variables is possible if the variables are discrete: • The frequency of one variable is subdivided by the other variable categories. • Generally a cross-tabulation table has: • Row percentages • Column percentages • Total percentages • Which one is better? DEPENDS on which variable is considered as independent.

  5. Cross tabulation

  6. Observed cases Total cases P= Contingency Table • A contingency table shows the conjoint distribution of two discrete variables • This distribution represents the probability of observing a case in each cell • Probability is calculated as:

  7. Chi-square Test for Independence • The Chi-square test for independence determines whether two variables are associated or not. H0:Two variables are independent H1:Two variables are not independent Chi-square test results are unstable if cell count is lower than 5

  8. Chi-Square Test Estimated cell Frequency Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Eij = estimated cell frequency Chi-Square statistic x² = chi-square statistics Oi = observed frequency in the ith cell Ei = expected frequency on the ith cell Degrees of Freedom d.f.=(R-1)(C-1)

  9. Awareness of Tire Manufacturer’s Brand Men Women Total Aware 50/39 10/21 60 Unaware 15/2125/1440 65 35 100

  10. X2 with 1 d.f. at .05 critical value = 3.84 Chi-Square Test: Differences Among Groups Example

  11. Chi-square Test for Independence • Under H0, the joint distribution is approximately distributed by the Chi-square distribution (2). Chi-square  Reject H0 3.84 2 22.16

  12. Differences Between Groups when Comparing Means • Ratio scaled dependent variables • t-test • When groups are small • When population standard deviation is unknown • z-test • When groups are large

  13. Null Hypothesis About Mean Differences Between Groups

  14. t-Test for Difference of Means

  15. X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means

  16. t-Test for Difference of Means

  17. t-Test for Difference of Means X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means.

  18. Pooled Estimate of the Standard Error

  19. Pooled Estimate of the Standard Error S12 = the variance of Group 1 S22 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2

  20. Pooled Estimate of the Standard Errort-test for the Difference of Means S12 = the variance of Group 1 S22 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2

  21. Degrees of Freedom • d.f. = n - k • where: • n = n1 + n2 • k = number of groups

  22. t-Test for Difference of Means Example

  23. Comparing Two Groups when Comparing Proportions • Percentage Comparisons • Sample Proportion - P • Population Proportion -

  24. Differences Between Two Groups when Comparing Proportions The hypothesis is: Ho: P1 = P2 may be restated as: Ho: P1 - P2 = 0

  25. Z-Test for Differences of Proportions or

  26. Z-Test for Differences of Proportions

  27. Z-Test for Differences of Proportions p1= sample portion of successes in Group 1 p2= sample portion of successes in Group 2 (p1 - p1)= hypothesized population proportion 1 minus hypothesized population proportion 1 minus Sp1-p2= pooled estimate of the standard errors of difference of proportions

  28. Z-Test for Differences of Proportions

  29. Z-Test for Differences of Proportions p= pooled estimate of proportion of success in a sample of both groups p= (1- p) or a pooled estimate of proportion of failures in a sample of both groups n1= sample size for group 1 n2= sample size for group 2

  30. Z-Test for Differences of Proportions

  31. Z-Test for Differences of Proportions

  32. A Z-Test for Differences of Proportions

  33. Analysis of Variance Hypothesis when comparing three groups m1 = m2 = m3

  34. Analysis of Variance F-Ratio

  35. Analysis of Variance Sum of Squares

  36. Analysis of Variance Sum of SquaresTotal

  37. pi= individual scores, i.e., the ith observation or test unit in the jth group pi= grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) Analysis of Variance Sum of Squares

  38. Analysis of Variance Sum of SquaresWithin

  39. pi= individual scores, i.e., the ith observation or test unit in the jth group pi= grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) Analysis of Variance Sum of SquaresWithin

  40. Analysis of Variance Sum of Squares Between

  41. = individual scores, i.e., the ith observation or test unit in the jth group = grand mean nj= number of all observations or test units in a group Analysis of Variance Sum of squares Between

  42. Analysis of Variance Mean Squares Between

  43. Analysis of Variance Mean Square Within

  44. Analysis of Variance F-Ratio

  45. A Test Market Experiment on Pricing Sales in Units (thousands) Regular Price $.99 130 118 87 84 X1=104.75 X=119.58 Reduced Price $.89 145 143 120 131 X2=134.75 Test Market A, B, or C Test Market D, E, or F Test Market G, H, or I Test Market J, K, or L Mean Grand Mean Cents-Off Coupon Regular Price 153 129 96 99 X1=119.25

  46. ANOVA Summary Table Source of Variation • Between groups • Sum of squares • SSbetween • Degrees of freedom • c-1 where c=number of groups • Mean squared-MSbetween • SSbetween/c-1

  47. ANOVA Summary Table Source of Variation • Within groups • Sum of squares • SSwithin • Degrees of freedom • cn-c where c=number of groups, n= number of observations in a group • Mean squared-MSwithin • SSwithin/cn-c

  48. ANOVA Summary Table Source of Variation • Total • Sum of Squares • SStotal • Degrees of Freedom • cn-1 where c=number of groups, n= number of observations in a group

More Related